Bounded Entire Functions Theorem Identification
The statement "Every bounded entire function is constant" is a fundamental result in complex analysis.
Understanding Key Terms
- Entire Function: A function that is analytic (holomorphic) over the entire complex plane ($\mathbb{C}$).
- Bounded Function: A function $f(z)$ is bounded if there exists a real number $M > 0$ such that $|f(z)| \le M$ for all $z$ in its domain.
Identifying the Theorem
The theorem directly linking these properties (boundedness and being entire) to the conclusion that the function must be constant is known as Liouville's theorem.
Theorem Statement: Liouville's Theorem
Liouville's theorem states that if a function $f(z)$ is both entire and bounded, then $f(z)$ must be a constant function.
Analysis of Options
- Morera's theorem: This theorem provides a condition for a continuous function to be analytic, based on the integral around closed paths being zero. It is the converse of Cauchy's integral theorem.
- Liouville's theorem: This is the correct theorem, stating that a bounded entire function is constant.
- Cauchy's theorem: This theorem states that the integral of an analytic function over a simple closed curve in a simply connected domain is zero.
- Cauchy's integral formula: This formula relates the value of an analytic function at a point inside a closed curve to the integral of the function over the curve.
Therefore, the correct identification is Liouville's theorem.
Important Questions from Mixed Topic (CUET PG)
- Who was the founder of Bolshevik Communist party?
- What is the key guide to statecraft in the realist tradition?
- Chronologically arrange the events in the Cold War period.
A. Berlin Wall is constructed
B. Communist China joins the UN
C. Soviet invasion of Czechoslovakia
D. Berlin Blockade
Choose the correct answer from the options given below: Morgenthau's principles of political realism are:
A. Politics is rooted in permanent and unchanging human nature which is basically self centred, self-regarding and self-interested
B. Politics is an autonomous sphere of action and cannot therefore be reduced to morals
C. International Politics is an arena of conflicting self-interests
D. The ethics of international relations is situational ethics which is very different from private morality
Choose the correct answer from the options given below:Who among the following political thinkers consider the anarchical self help system to be a compelling factor for States to maximise their relative power positions?
